San Francisco Crime and Topography

Visitors and Residents of San Francisco are familiar with the hilly terrain and roads thatclimb into the clouds. But does this topography have an effect on the likelihood of car break-ins? This regression analysis seeks to provide insights into this question.

That is one steep street!

Research Question

My research question is: "What is the relationship between topography and car break-ins in San Francisco?

Background

This analysis will focus on the link between elevation and “hilliness” in determining the likelihood of car break-ins in San Francisco. Both terrain and motor vehicle crimes are ubiquitous when discussing living or visiting San Francisco. This relationship has been coined by the phrase - “Crime Doesn’t Climb”.

In April 2021, Young-An Kim & James C. Wo published Topography and crime in place: The effects of elevation, slope, and betweenness in San Francisco street segments. Their study provides a robust regression analysis on the effects of elevation, slope, and “hilliness” on crime, controlling for socio-demographic characteristics Kim and Wo (2021). My analysis will focus only on car break-ins rather than all crime reports, as I believe that these crimes will have an even more pronounced relationship with topography.

Data Collection

In order to collect data, we first need to identify our key variables.

Important Variables

• Dependent variable: Crime (specifically car break-ins)

• Independent variables: Elevation and slope

• Control variables: Median income by census block

When discussing topography, both elevation and “hilliness,” or slope, are necessary for inclusion. This is because it more accurately captures the effect of local level topography, which is supported by Kim & Wo.

In any econometric analysis, it is vital to control for socio-economic variables. In this case, it could be that higher elevations in the city are more affluent areas, which may have an impact on crime. Thus, we want to include median income as a control variable.

Here is the regression equation:

\[MotorVehicleTheft_i = \beta_0 + \beta_1Elevation_i + \beta_2Slope_i + \beta_3MedianIncome_i + u_i\]

Data Sources

Crime and elevation data for this analysis were retrieved from the San Francisco Open Data Portal.

• Crime - All crime reports from 2018-01-01 to 2021-11-04.

• Elevation contours - 5 ft. elevation contours, which were also used to derive slope.

• Median income: Retrieved through the tidycensus package via the US Census Bureau.

Analysis Plan

The analysis plan steps to address this research question are as follows:

1. Identify question*
2. Select key variables (based on existing literature)*
3. Collect data*
4. Merge data
5. Visualize simple relationships
6. Test OLS assumptions
7. Conduct regression analysis
8. Interpret Results

Completed in previous sections*

Merge Data

The following code chunk demonstrates how the crime, elevation, slope, and income datasets were merged.

Show code

# Find index of nearest contour to each crime
elev <- st_nearest_feature(x = crimes, y = contours)

# Add elevation and binary slope columns
crimes <- crimes %>% 
  st_join(y = census_geom, join = st_within, left = TRUE) %>% 
  mutate(elev = contours[elev,]$elevation) %>% 
  rename(median_income = estimate) %>% 
  select(date_incid, slope, median_income, elev, geometry)

Visualize simple relationships

The following plots show the simple relationships between count of car break ins and the three independent variables (elevation, slope, and median income). We see there is a negative correlation between elevation and crime. It is possible that this relationship is not linear, so this variable may fit better if we take the natural log. Since slope in this analysis is a binary variable, we simply see that there are more crimes in areas that are not designated high slope. Lastly, we observe a weak negative correlation between median income and car break-ins.

Show code

# Group by income only
income_summary <- crimes %>% 
  st_drop_geometry() %>% 
  group_by(median_income) %>% 
  summarize(count = n())

income_plot = ggplot(data = income_summary, aes(x = median_income, y = count)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE) +
  theme_classic() +
  labs(title = "Crime and Median Income",
       x = "Median Income (USD)",
       y = "Number of Break-Ins")

# Group by elevation
elev_summary <- crimes %>% 
  st_drop_geometry() %>% 
  group_by(elev) %>% 
  summarize(count = n())

elev_plot <- ggplot(data = elev_summary, aes(x = elev, y = count)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE) +
  theme_classic() +
  labs(title = "Crime and Elevation",
       x = "Elevation (feet)",
       y = "Number of Break-Ins")

# Group by slope
slope_summary <- crimes %>% 
  st_drop_geometry() %>% 
  group_by(slope) %>% 
  summarize(count = n())

slope_plot <- ggplot(data = slope_summary, aes(x = slope, y = count)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE) +
  theme_classic() +
  labs(title = "Crime and Slope",
       x = "Slope (percent)",
       y = "Number of Break-Ins")

elev_plot + (slope_plot / income_plot)

The following plots show crimes over the time period of our data. We can see that there is a slight upward trend to our crime data.

Show code

# Group by all three variables
crimes_summary <- crimes %>% 
  st_drop_geometry() %>% 
  group_by(slope, median_income, elev) %>% 
  summarize(count = n())

# Group by date
crimes_ts <- crimes %>% 
  st_drop_geometry() %>% 
  group_by(date_incid) %>% 
  summarize(count = n())

ts_plot <- ggplot(data = crimes_ts, aes(x = date_incid, y = count)) +
  geom_point() +
  geom_smooth(method = "lm") +
  theme_classic() +
  labs(title = "Daily Crime 2018-Present",
       x = "Date (daily)",
       y = "Number of Break-Ins")

crimes_monthly <- crimes_ts %>% 
  mutate(month = lubridate::floor_date(date_incid, "month")) %>% 
  group_by(month) %>% 
  summarize(monthly_sum = sum(count)) %>% 
  filter(month != "2021-11-01")

monthly_plot <- ggplot(data = crimes_monthly, aes(x = month, y = monthly_sum)) +
  geom_point() +
  geom_line() + 
  #geom_smooth(method = "lm") +
  theme_classic() +
  labs(title = "Monthly Crime 2018-Present",
       x = "Date (monthly)",
       y = "Number of Break-Ins")

ts_plot + monthly_plot

Show code

bbox <- st_bbox(census_geom)

library(tmap)
tmap_mode("view")
tm_shape(crimes)+
  tm_dots("elev")

Show code

#elev_map <- ggplot(data = crimes) +
#  geom_sf(aes(color = elev)) +
#  theme_minimal() +
#  theme(axis.text = element_blank(),
#        legend.position = "bottom",
#        legend.title = element_blank()) +
#  labs(title = "Crime by Elevation")

#slope_map <- ggplot(data = crimes) +
#  geom_sf(aes(color = slope)) +
#  theme_minimal() +
#  theme(axis.text = element_blank(),
#        legend.position = "bottom",
#        legend.title = element_blank()) +
#  labs(title = "Crime by Slope")

#income_map <- ggplot(data = crimes) +
#  geom_sf(aes(color = median_income)) +
#  theme_minimal() +
#  theme(axis.text = element_blank(),
#        legend.position = "bottom",
#        legend.title = element_blank()) +
#  labs(title = "Crime by Median Income")

#elev_map #+ (slope_map / income_map)

Regression Analysis

Show code

model <- lm(formula = count ~ elev + slope + median_income, data = crimes_summary)

#model %>% 
#  summary() %>% 
#  xtable() %>% 
#  kable()

summ(model) 

Observations	4391 (23 missing obs. deleted)
Dependent variable	count
Type	OLS linear regression

F(3,4387)	151.47
R²	0.09
Adj. R²	0.09

	Est.	S.E.	t val.	p
(Intercept)	10.15	0.36	28.27	0.00
elev	-0.01	0.00	-14.89	0.00
slope	-0.15	0.02	-6.88	0.00
median_income	-0.00	0.00	-3.43	0.00
Standard errors: OLS

Transformation

Show code

model_transformed <- lm(formula = log(count) ~ elev + slope + median_income, data = crimes_summary)

summ(model_transformed)

Observations	4391 (23 missing obs. deleted)
Dependent variable	log(count)
Type	OLS linear regression

F(3,4387)	247.60
R²	0.14
Adj. R²	0.14

	Est.	S.E.	t val.	p
(Intercept)	1.88	0.04	48.15	0.00
elev	-0.00	0.00	-18.72	0.00
slope	-0.02	0.00	-9.09	0.00
median_income	-0.00	0.00	-4.69	0.00
Standard errors: OLS

The coefficients above can be interpreted as the effect of the independent variables on car break-ins.

Test OLS Assumptions

Recall our four key assumptions for OLS:

1. The population relationship is linear in parameters with an additive disturbance.

2. Our \(X\) variables are exogenous, i.e., \(\mathop{\boldsymbol{E}}\left[ u \mid X \right] = 0\).

3. The \(X\) variables have variation.

4. The population disturbances \(u_i\) are independently and identically distributed as normal random variables with mean zero \(\left( \mathop{\boldsymbol{E}}\left[ u \right] = 0 \right)\) and variance \(\sigma^2\) (i.e., \(\mathop{\boldsymbol{E}}\left[ u^2 \right] = \sigma^2\))

We assume that assumptions 1 and 2 hold. Assumption 3 holds because we see that the \(x\) variables have variation in the plots above. Below, the residuals are generated from the main regression and are used to assess the three components of assumption #4.

Show code

crime_lm <- crimes_summary %>% 
  add_predictions(model = lm(count ~ elev + slope + median_income, data = crimes_summary)) %>% 
  mutate(res = count - pred)

ggplot(data = crime_lm, aes(x = res)) +
  geom_histogram(binwidth = 5, fill = "darkblue", col = "black") + 
  labs(title = "Residual Plot",
       x = "Residuals",
       y = "Count") +
  theme_classic()

The residuals do not appear to be normally distributed. There is a long right tail with high outliers. This indicates that the regression analysis is systematically under-predicting counts of car break-ins.

Show code

ggplot(data = crime_lm, aes(sample = res)) +
  geom_qq() +
  geom_qq_line() +
  theme_classic() +
  labs(title = "QQ Plot",
       x = "Theoretical Normal Distribution",
       y = "Analysis Distribution")

The qq plot shows that the distribution is mostly normally distributed for values up to approximately 2. This suggests that the model is not normally distributed for extreme high values.

Show code

x <- ggplot(data = crime_lm, aes(x = count, y = res)) +
  geom_point() + theme_classic() +
  labs(title = "Residuals vs Dependent (Break-Ins)",
       x = "Number of Break-Ins",
       y = "Residual Error")

y1 <- ggplot(data = crime_lm, aes(x = elev, y = res)) +
  geom_point() + theme_classic() + 
  labs(title = "Residuals vs. Elevation",
       x = "Elevation (feet)",
       y = "Residual Error")

y2 <- ggplot(data = crime_lm, aes(x = slope, y = res)) +
  geom_point() + theme_classic() + 
  labs(title = "Residuals vs. Slope",
       x = "Slope (percent)",
       y = "Residual Error")

y3 <- ggplot(data = crime_lm, aes(x = median_income, y = res)) +
  geom_point() + theme_classic() + 
  labs(title = "Residuals vs. Income",
       x = "Median Income",
       y = "Residual Error")

(x + y1) / (y2 + y3)

Since assumptions 1, 2, and 3 appear to be satisfied, using OLS in this case is an unbiased estimator. However, residual plots indicate that assumption 4 may not be satisfied and therefore OLS may not be the estimator with lowest variance. Thus, an alternative estimator to OLS may provide estimates of the true population parameters with less variance.

Results